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Research on Teacher Education in Mathematics -- through collaborative inquiry-based processes including teachers and students Barbara Jaworski APEC - TSUKUBA International Conference – 2015 Mathematics Education Centre

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Structure Collaboration in research and teaching through inquiry What does it mean to teach mathematics (or science)? Inquiry in the classroom The role of the teacher Inquiry in theory Collaborative inquiry between teachers and didacticians Video example from a research project Collaborative developmental research Tsukuba APEC Conference Image : understand-the-music-of-their-web/

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Co-learning inquiry Tsukuba APEC Conference In a co-learning agreement, researchers and practitioners are both participants in processes of education and systems of schooling. Both are engaged in action and reflection. By working together, each might learn something about the world of the other. Of equal importance, however, each may learn something more about his or her own world and its connections to institutions and schooling. (Wagner, 1997, p. 16) Image: The DanceImage: The Dance. Designed and folded by H.T.Quyet from 1 uncut right triangle – half of a square size 34×34 cm. (via ORI_Q)ORI_Q

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Central concepts Mathematics Research Inquiry Learning Teaching Development Image from The Uplands Teaching and Learning Blog Learning Mathematics through Inquiry Learning Mathematics Teaching through Inquiry Collaboration in learning and teaching Communities of practice and inquiry Critical Alignment Teachers and didacticians as researchers Tsukuba APEC Conference

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What is teaching? An action might be described as ‘teaching’ if, first, it aims to bring about learning, second, it takes account of where the learner is at, and, third it has regard for the nature of what has to be learnt. (Pring, 2000, p. 23) For example, the teaching of a particular concept in mathematics can be understood only within a broader picture of what it means to think mathematically, and its significance and value can be understood only within the wider evaluation of the mathematics programme. (Pring, 2000, p.27) [not attending to this] is to accept a limited and impoverished understanding of teaching. (Pring, 2004, p. 22) Tsukuba APEC Conference Image is the Harriss spiral is constructed using a simple process of dividing rectangles

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… a moral practice Teaching … is more than a set of specific actions in which a particular person is helped to learn this or that. It is an activity in which the teacher is sharing in a moral enterprise, namely the initiation of young people into a worthwhile way of seeing the world … … there can be no avoidance of that essentially moral judgement of the teacher over what is worth learning and what are the worthwhile ways of pursuing it. (Pring, 2004, p. 18) Tsukuba APEC Conference Image a 22ft tall Sierpinski tetrahedron Designed by Gwen Fisher, engineered by Paul Brown.Sierpinski tetrahedron

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Teaching for conceptual understanding The initiation of young people into a worthwhile way of seeing the world – the broader picture. The moral judgement of the teacher over what is worth learning and what are the worthwhile ways of pursuing it. What is worthwhile in mathematics? to do mathematics to use mathematics to apply mathematics Requires mathematical understanding Tsukuba APEC Conference Image is A crocheted model of the hyperbolic plane. Photograph: Daina Taimin

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Bringing inquiry into the classroom Tsukuba APEC Conference A demonstration of the mathematical principles of the original Forth Bridge in Scotland performed at Imperial College in The central 'weight' is Kaichi Watanabe, one of the first Japanese engineers to study in the UK, while Sir John Fowler and Benjamin Baker provide the supports.

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Tsukuba APEC Conference Inquiry in the classroom Ask questions and seek answers Recognise problems and seek solutions Wonder Investigate Explore Imagine Look critically Reason Discuss STUDENTS ! WHO?

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Inquiry-based tasks Inspire involvement Provide easy access to mathematical ideas Enable everyone to make a start Provide opportunity to ask questions, solve problems, imagine, explore, investigate … Encourage discussion and reasoning, diverse directions and levels of thinking, fluidity and flexibilility Encourage student centrality/ownership in/of the mathematics Promote serious mathematical thinking Tsukuba APEC Conference

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What kinds of tasks am I talking about? Tsukuba APEC Conference

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Some inquiry-based problems/tasks A. Take any natural number and express it as a sum in as many ways as possible (e.g. 10 = 2+8 = = ) Multiply the numbers of any sum and find the largest product. Generalise? B. You have a number of balls in a bag, some red and some white. What is the least number of red balls for which the probability of drawing two red balls in succession (without replacement) is more than 1/3? Generalise? Tsukuba APEC Conference C. Can you find any plane shape for which the area is numerically equal to its perimeter? Generalise?

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How tall a mirror must you buy if you want to be able to see your full vertical image? Tsukuba APEC Conference Grade 1 One student faces a mirror holding a stick (against his stomach). This student directs another, who, using a whiteboard marker, marks the mirror image the first one sees. Compare the original stick with the marks on the mirror. Try different distances from the mirror. Grade 2 One student holds a geometric figure (against the stomach) and explains to another student how to draw (on the mirror) the mirror image he sees. Compare. Grade 3 Measure yourself in centimetres. Measure your mirror image in centimetres. Draw yourself seeing yourself in a mirror.

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Grade 6 Measure yourself and your mirror image.Draw yourself (simplified) looking ina mirror with the correct ratios (and angles) in your drawing. Grade 7 Draw model of a figure and an eye and the mirror image the eye sees (keep the eye and the figure at the same distance from the mirror?). Describe lengths and angles. What do you see? Grade 8 Hold a cube and go close to the mirror. Draw on the lines of the cube on the mirror. What do you see? Grade 11 How tall a mirror must you buy if you want to be able to see your full vertical image? Justify your conclusion; try with objects with different distances from mirror; describe ratios in your model Grade 12 How tall a mirror must you buy if you want to be able to see your full vertical image? Justify your conclusion; try with objects with different distances from mirror; describe ratios in your model use the cosine rule to derive the height of the actual figure when the height of the mirror image is known Grade 13 Draw yourself and a mirror in a three dimensional vector space. Tsukuba APEC Conference

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Topic-focused tasks Given the function f(x) = x 2 – 3x + 4, x is a real number sketch the function on a pair of axes. a)Find the equation of a line that crosses this curve where x=1 and x=2 b)Find the equation of a line of gradient 3 that crosses the curve twice c)Find the equation of a line of gradient -3 that does not cross the curve Tsukuba APEC Conference Use sliders in GeoGebra to determine which of the graphs below could represent the function y=ax 4 +bx 3 +cx 2 +dx+e Here a, b, c, d and e are real numbers, and a≠0. Explain Jaworski & Matthews, 2011

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What is the teacher doing … … when students work on inquiry-based tasks? Circulating and listening Asking, and encouraging students to ask questions Encouraging dialogue and/or debate Fostering reasoning Prompting and challenging Tsukuba APEC Conference What is the teacher’s role?

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Tsukuba APEC Conference Some theoretical ideas Conjecturing atmosphere (Mason, Burton & Stacey, 1982) Encouraging students to conjecture and test their conjectures – leading to generalisation, abstraction and proof in mathematics Teaching triad (Jaworski, 1994) Management of learning Sensitivity to students Mathematical challenge HARMONY – balance between sensitivity and challenge (Potari & Jaworski, 2002) ML SSMC

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Tsukuba APEC Conference Inquiry in classroom mathematics Students engaging in mathematical tasks that are inquiry- based allows Multiple directions of inquiry Making, testing and justifying mathematical conjectures Multiple levels of engagement Mutual engagement and support Differing degrees of challenge Harmony in balancing sensitivity and challenge Acceptance of and respect for difference

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Tsukuba APEC Conference Inquiry in mathematics teaching Involves teachers-as-inquirers exploring the kinds of tasks that engage students and promote mathematical inquiry ways of organising the classroom that enable inquiry activity with all its attributes The many issues and tensions that arise related to the classroom, school, parents, educational system, society and politics. and reflecting on what occurs in the classroom with feedback to future action

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An inquiry cycle Plan for teaching Act and observe Reflect and analyse Feedback to future planning Tsukuba APEC Conference e.g.,Design a particular classroom task Use the task in the classroom and observe what happens -- gather data Reflect on what occurred -- analyse the data Use what has been learned through observation and analysis to redesign the task Action research Design research Iterative and systematic (Jaworski, 2004)

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Inquiry in two layers Inquiry in students’ mathematical activity in the classroom Inquiry in teachers’ exploration of classroom approaches Tsukuba APEC Conference

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Two communities of inquiry Tsukuba APEC Conference Inquiry in classroom mathematics – -- students (& teacher) Inquiry in planning for the classroom -- teachers (& teacher educators -- didacticians)

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Theoretical background: Community of Practice From Vygotskian theory (e.g., Vygotsky, 1978, Wertsch, 1991) Learning as participation in social action mediated by tools and signs From Wenger’s “Community of Practice” (CoP ) (Wenger, 1998) Mutual engagement; Joint enterprise; Shared repertoire Wenger’s notion of belonging to a CoP, involves Engagement; Imagination; Alignment BUT Alignment can result in perpetuation of unhelpful practices (e.g., instrumental learning and teaching in mathematics (e.g., Skemp 1985) Tsukuba APEC Conference

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From community of practice … Involves Engagement Imagination and Alignment Perpetuation of established practices (Brown and McIntyre, 1993) Tsukuba APEC Conference 2015 Inquiry – new kinds of engagement Critical alignment to Community of Inquiry 24

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Critical Alignment in Practice We look critically at our practice, while we engage and align with it Ask questions about what we are doing and why Reveal and question implicit assumptions and expectations Try out innovative approaches to explore alternative ways of doing and being to achieve our fundamental goals Tsukuba APEC Conference

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Tsukuba APEC Conference Creating a Community of Inquiry Within a community of practice Ask questions and seek answers Recognise problems and seek solutions Wonder, imagine, invent, explore … Look critically: ‘critical alignment’ leads to ‘metaknowing’ (Jaworski, 2006; 2008; Wells, 1999) From inquiry as a tool To inquiry as a way of being in practice Establishing an inquiry identity Community of inquiry transformation

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An inquiry identity Having an inquiry identity is more than (just) doing the occasional inquiry-based task in the classroom. It involves being an inquirer in all our professional activity designing and using inquiry-based tasks with students encouraging students to ask their own questions and explore aspects of mathematics for themselves Questioning our own approaches to teaching Dealing with issues in teaching, in collaboration with colleagues Exploring (new) approaches to teaching, with our colleagues Working with didacticians to develop teaching. Tsukuba APEC Conference

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Collaborative inquiry Two groups of people have important knowledge to bring to teaching development Teachers Teacher educators/researchers (didacticians) Together, these kinds of knowledge can be powerful in the developmental process. Tsukuba APEC Conference

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Tsukuba APEC Conference Didacticians’ knowledge of theory, research and systems Knowledge shared by didacticians and teachers: e.g. mathematics pedagogy didactics Teachers’ knowledge of students and schools Systemic and cultural settings and boundaries within which learning and teaching are located Teachers How is knowledge distributed? Didacticians

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Tsukuba APEC Conference Inquiry in three layers Inquiry in students’ mathematical activity in the classroom Inquiry in teachers’ exploration of classroom approaches Inquiry in addressing questions and issues to do with teaching and how it can develop to promote mathematics learning. Inquiry in mathematics Inquiry in mathematics teaching Inquiry in research into learning and teaching mathematics

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Developmental research in Norway A project involving a team of didacticians and teachers in 8 schools 3 phases over 3 school years Based in inquiry at three levels Workshops in the university led by didacticians Developmental work in schools led by teachers Video compilation Video compilation Tsukuba APEC Conference

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Developmental research/inquiry Students engage in inquiry in mathematics in the classroom. Teacher-researchers together explore ways of engaging with students using inquiry-based tasks to promote learning Didactician-researchers encourage teachers collaborative inquiry and research the processes and practices involved. Lesson Study fits this model Tsukuba APEC Conference

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Thank you for listening Tsukuba APEC Conference

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References Brown, S., & McIntyre, D. (1993). Making sense of teaching. Buckingham, UK: Open University Press. Eraut, M. (1995). Sch¨on shock: A case for reframing reflection-in-action? Teachers andTeaching: Theory and Practice, 1(1), 9–22. Jaworski, B. (1988). ‘Is’ versus ‘seeing as’. In D. Pimm (Ed.). Mathematics Teachers and Children. London: Hodder and Stoughton. Jaworski, B. (1994). Investigating Mathematics Teaching. London: Falmer Press Jaworski, B. (2003) Research practice into/influencing mathematics teaching and learning development: towards a theoretical framework based on co-learning partnerships. Educational Studies in Mathematics 54, 2-3, Jaworski, B. (2004). Insiders and outsiders in mathematics teaching development: the design and study of classroom activity. Research in Mathematics Education, 6, Jaworski B. (2006). Theory and practice in mathematics teaching development: Critical inquiry as a mode of learning in teaching. Journal of Mathematics Teacher Education, 9(2), Jaworski, B., Fuglestad, A. B., Bjuland, R., Breiteig, T., Goodchild, S., & Grevholm, B. (Eds.), (2007). Learning communities in mathematics. Bergen, Norway: Caspar. Jaworski, B., Goodchild, S., Daland, E., & Eriksen S. (2011). Mediating mathematics teaching development and pu-pils’ mathematics learning: the life cycle of a task. In O. Zaslavsky & P. Sullivan (Eds.), Constructing knowledge for teaching secondary mathematics: Tasks to enhance prospective and practicing teacher learning. New York: Springer. Jaworski, B., Fuglestad, A. B., Bjuland, R., Breiteig, T., Goodchild, S., & Grevholm, B. (Eds.). (2007). Learning communities in mathematics. Bergen, Norway: Caspar. Jaworski, B. & Matthews, J. (2011) Developing teaching of mathematics to first year engineering students. Teaching Mathematics and Its Applications 30(4): Tsukuba APEC Conference

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Mason, J., Burton, L. & Stacey, K. (1982). Thinking Mathematically. London: Addison Wesley McIntyre, D. (1993). ‘Theory, Theorizing and Reflection in Initial Teacher Education.’ In J. Calderhead and P. Gates (Eds.), Conceptualising Reflection in Teacher Development (pp ). London, Falmer. Potari, D. & Jaworski, B. (2002). Tackling complexity in mathematics teaching. Journal of Mathematics Teacher Education, 5, 4, Pring, R (2000). The Philosophy of Educational Research. London: Continuum. Pring, R. (2004). Philosophy of Education. London: Continuum. Open University (1985). Working Mathematically with Low Attainers: Centre for Mathematics Education Research Study. Milton Keynes: Open University. Rowland, T., Huckstep, P. & Thwaites, A. (2005). Elementary Teachers Mathematics Subject Knowledge. Journal of Mathematics Teacher Education, 8, 3, Skemp, R. (1989). Mathematics in the primary school. London: Routledge. Smith, A. (2004). Making mathematics count. The report of Professor Adrian Smith’s inquiry into post-14 mathematics education. London: The Stationery Office. Stenhouse, L. (1984). Evaluating curriculum evaluation. In C. Adelman (Ed.). The Politics and Ethics of Education. London: Croom Helm. Vygotsky, L. (1978). Mind in society. Cambridge, MA: Harvard University Press. Wagner, J. (1997). The unavoidable intervention of educational research: A framework for reconsidering research-practitioner cooperation. Educational Researcher, 26(7), 13–22. Wells, G. (1999). Dialogic inquiry: Toward a sociocultural practice and theory of education. Cambridge, UK: Cambridge University Press. Wenger, E. (1998). Communities of practice: Learning, meaning and identity. Cambridge, UK: Cambridge University Press. Wertsch, J. V. (1991). Voices of the mind. Cambridge, MA: Harvard University Press. Tsukuba APEC Conference

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Tsukuba APEC Conference

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Turning “I can’t” into “I can and I did” (Open University, 1985) A teacher set her class the following task: If a number of circles intersect in a plane, how many regions can be created? What is the maximum number? Pupils used hoops on the ground or drew circles on paper or on the whiteboard 1 circle 1 region 2 circles, 3 regions 3 circles, 7 regions Conjecture: the number of regions for n circles is [2 n – 1] Test: when n=4 [2 n – 1] = [2 4 – 1] = 15 4 circles, 13 regions 5 … Tsukuba APEC Conference ? ? ?

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Tsukuba APEC Conference Mary had done something different … She brought it to the teacher. The teacher discussed with her what she was finding from this investigation. Then finally the teacher said: “You’re doing something different from everyone else, Mary. Don’t worry, that’s fine. Just ignore what everyone else is doing” Opportunity for all, allows one to be different

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What shape is it? (Jaworski, 1988) Look at the figure here. What is it? What shape is it? A class of 12 year olds had been asked by their teacher to name the above shape, which he had drawn on the board. Someone said that it was a trapezium. Some students agreed with this, but not all. The teacher said, ‘If you think it’s not a trapezium then what is it?’ Michael said, tentatively, ‘It’s a square …’ There were murmurings, giggles, ‘a square’?! … But Michael went on ‘… sort of flat.’ Tsukuba APEC Conference

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Contingency (Rowland, Huckstep & Thwaites, 2005) The teacher looked puzzled, as if he could not see a square either. He invited Michael to come out to the board and explain his square. Michael did. He indicated that you had to be looking down on the square – as if it were on your book, only tilted. He moved his hand to illustrate. ‘Oh’ said the teacher. ‘Oh, I think I see what you mean … does anyone else see what he means?’ There were more murmurings, puzzled looks, tentative nods. Then the teacher drew … Oooooh yes (!) said the students and there were nods around the class. Tsukuba APEC Conference Responding in the moment to something different (inquiring) -- and using it for a good outcome for all

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